Data-Driven Reduction of Intrisically Nonlinear Dynamics to Spectral Submanifolds: Theory and Applications

Machine learning has been a major development in applied science and engineering, with impressive success stories in static learning environments like image, pattern, and speech recognition. Yet the modeling of dynamical phenomena—such as nonlinear vibrations of solids and transitions in fluids—remains a challenge for classic machine learning. Indeed, neural net models for nonlinear dynamics tend to be complex, uninterpretable and unreliable outside of their training range.



In this talk, I discuss a recent dynamical systems alternative to neural networks in the data-driven reduced-order modeling of nonlinear phenomena. Specifically, I show that the concept of spectral submanifolds (SSMs) provides very low-dimensional attractors in a large family of physical problems ranging from predicting wing oscillations to controlling soft robots. A data-driven identification of the reduced dynamics on these SSMs gives a rigorous way to construct accurate and predictive reduced-order models in solid and fluid mechanics without the use of governing equations. I illustrate this on problems that include reduced-order modeling of fluid sloshing in a tank, identification of matrerial nonlinearities of hydrogels and model-predictive control of soft robots.